Iwasawa Continued Fractions: Higher-Dimensional Geodesic Generalizations with Proven Convergence — E8 Intelligence Research

Abstract

FINDING: Iwasawa continued fractions generalize classical continued fractions to higher-dimensional hyperbolic spaces via geodesic coding on SL(n,R)/SO(n), with proven convergence and ergodicity for complex, quaternionic, octonionic, and Heisenberg variants. | MATH: Classical CF: \(x = [a_0; a_1, a_2, \dots]\), \(a_i \in \mathbb{Z}\). Iwasawa CF: decomposition \(SL(n,\mathbb{R}) = NAK\) (unipotent × diagonal × orthogonal), geodesic flow on symmetric space \(X = SL(n,\mathbb{R})/SO(n)\) coded by \(N\)-part (Iwasawa coordinates). Ergodic theorem: for almost all \(x\), the Iwasawa CF converges; invariant measure is the Haar measure on \(SL(n,\mathbb{R})\). Key constants: for \(n=2\), reduces to classical Gauss map with Gauss–Kuzmin–Wirsing constant \(\lambda \approx 0.30366\); for \(n>2\), Lyapunov spectra relate to \(\log 2, \log 3, \dots\) (entropy of geodesic flow = \(n-1\) in curvature-normalized units). | CONNECTION: The \(A\)-part of Iwasawa decomposition is diagonal with entries \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com